Winger Ryan Flynn is available again for the Blades after recovering from illness but was left out by manager Danny Wilson. Bury include midfielder Giles Coke after his loan move from Sheffield Wednesday.
Richard Cresswell broke the Bury resistance on 54 minutes, breaking the offside trap after good work from Kevin McDonald.
Further goals from Ched Evans and Lee Williamson put the Blades in complete control.
With city rivals Wednesday losing to Charlton, it means United pulled four points clear of their neighbours.
For full reports and reaction read Sports Monday, free inside Monday’s Yorkshire Post newspaper.